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1 Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב INTRODUCTION

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Instructors Dr. Liad Blumrosen ד"ר ליעד בלומרוזן –Department of economics, huji. Dr. Michael Schapira ד"ר מיכאל שפירא –School of computer science and engineering, huji. Office hours: by appointment.

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Course requirements Attend (essentially all) classes. Solve 3-4 problem sets. –The final problem set might be slightly bigger. Problem sets grade is 100% of the final grade. –No exam, no home exam.

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Computer science and economics ?!? Today: –Introduction and examples –Game theory 1.0.1.

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Classic computer science What a single computer can compute?

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Classic Economics Analyzing the interaction between humans, firms, etc.

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New computational environments Properties: – Large-scale systems, belong to various economic entities. – Participants are individuals/firms with different goals. – Participants have private information. – Rapid changes in users behavior. Electronic marketsInformation providers Social networksP2P networks Internet Mobile and apps

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Algorithmic game theory Which tools can we use for analyzing such environments? Interactions between computers, owned by different economic entities and different goals. algorithmic game theoryNew tools should be developed: algorithmic game theory The theory borrows a lot from each field.

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What tools should we use? “Classic” CS Not handling, eg: Incentives Asymmetric information Participation constraints Economics / Game theory Not handling, eg: Tractability Approximation Various objectives Algorithmic Game Theory: + Design & evaluate systems with selfish agents. + Real need from the industry.

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Few examples

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Example 1: Single-Item Auctions 2 nd -price auction Buyers submit bids Highest bid wins Winner pays the 2 nd - highest bid In which auction would you bid higher? How do people behave in such auctions? Which one earns greater revenue for the seller? 1 st -price auction Buyers submit bids Highest bid wins Winner pays his own bid Say that you need to sell a single (indivisible) item to a set of bidders. How can you do that?

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Example 1: Single-Item Auctions Auctions are part of the mechanism design literature. Mechanism design: economists as engineers. Design markets with selfish agent to achieve some desired goals. –Relation to computer science is straightforward. –Once a niche field in economics, now mainstream. See this year’s Nobel prize (+ 2007, 1994)

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Example 2: Sponsored-search auctions Bla Search resultsAdvertisements

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Example 2: Sponsored-search auctions A real system: A simple interface short response time robustness Selfish parties: Google vs. Yahoo vs. MSN Users Advertisers Economic challenges, eg: Which auction to use? Private info – how much advertisers will pay? Click Fraud Attract new advertisers payments per impression/click/action

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Example 3: FCC spectrum auctions Multi-billion dollar auctions. Preferences for bundles of frequencies (Combinatorial auctions) : Consecutive geographic areas. Overlaps, already owned spectrum. Sophisticated bidders –At&t, Verizon, Google. –Again, asymmetric information. Bottleneck: communication.

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Example 4: selfish routing 16 Many cars try to minimize driving time. All know the traffic congestion (גלגלצ, WAZE)

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Externalities and equilibria 17 Negative externalities: my driving time increases as more drivers take the same route. In “equilibrium”: no driver wants to change his chosen route. Or alternatively: –Equilibrium: for each driver, all routes have the same driving time. (Otherwise the driver will switch to another route…)

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Efficiency, equilibrium. 18 Our question: are equilibria socially efficient? –Would it be better for the society if someone told each driver how to drive? We would like to compare: –The socially-efficient outcome. What would happen if a benevolent planner controlled traffic. –The equilibrium outcome. What happens in real life.

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Network 1 19 Socially efficient outcome: splitting traffic equally –expected driving time: ½*1+½*1/2=3/4 –Exercise: prove this is efficient. The only equilibrium: everyone use lower edge. –Otherwise, if someone chooses upper link, the cost in the lower link is less than 1. –Expected cost: 1*1=1 C(n)=n C(n)=1 (million) c(n) – the cost (driving time) to users when n users are using this road. Assume that a flow of 1 (million) users use this network. ST

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Network 1 20 Conclusion: –Letting people choose paths incurs a cost –“price of anarchy” The immediate question: if we have a ratio of 75% for this small network, can it be much higher in more complex networks? Which networks? C(n)=n C(n)=1 (million) ST

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Network 2 21 In equilibrium: half of the traffic uses upper route half uses lower route. Expected cost: ½*(1/2+1)+1/2*(1+1/2)=1.5 c(n)=n c(n)=1 ST c(n)=n c(n)=1

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Network 3 22 The only equilibrium in this graph: everyone uses the s v w t route. –Expected cost: 1+1=2 Building new highways reduces social welfare!? c(n)=n c(n)=1 ST v W c(n)=n c(n)=1 c(n)=0 Now a new highway was constructed!

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Braess’s Paradox 23 This example is known as the Braess’s Paradox: sometimes destroying roads can be beneficial for society. The immediate question: how can we choose which roads to build or destroy? c(n)=n c(n)=1 ST v W c(n)=n c(n)=1 c(n)=0 Now a new highway was constructed!

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Example 5: Internet Routing Establish routes between the smaller networks that make up the Internet Currently handled by the Border Gateway Protocol (BGP). AT&T Qwest Comcast Level3

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Why is Internet Routing Hard? Not shortest-paths routing!!! AT&T Qwest Comcast Level3 My link to UUNET is for backup purposes only. Load-balance my outgoing traffic. Always choose shortest paths. Avoid routes through AT&T if at all possible.

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BGP Dynamics 1 2 d 2, I’m available 1, my route is 2d 1, I’m available Prefer routes through 2 Prefer routes through 1

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Two Important Desiderata BGP safety – Guaranteeing convergence to a stable routing state. Compliant behaviour. –Guaranteeing that nodes (ASes) adhere to the protocol.

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We saw examples for modern systems that raise many interesting questions in algorithmic game theory. Next: a quick introduction to game theory Outline: –What is a game? –Dominant strategy equilibrium –Nash equilibrium (pure and mixed)

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29 Game Theory Game theory involves the study of strategic situations Portrays complex strategic situations in a highly simplified and stylized setting –Strategic situations: my outcome depends not only on my action, but also on the actions of the others. A central concept: rationality –A complex concept. Many definitions. –One possible definition: Agents act to maximize their own utility subject to the information the have and the actions they can take.

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Applications Economics –Essentially everywhere Business –Pricing strategies, advertising, financial markets… Computer science –Analysis and design of large systems, internet, e- commerce. Biology –Evolution, signaling, … Political Science –Voting, social choice, fair division… Law –Resolutions of disputes, regulation, bargaining… …

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31 Game Theory: Elements All games have three elements –players –strategies –payoffs Games may be cooperative or noncooperative –In this course, noncooperative games.

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32 Let’s see some examples….

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Example 1: “chicken” Chicken!!! SwerveStraight Swerve 0, 0-1, 1 Straight 1, -1-10,-10

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Example 2: Prisoner’s Dilemma Two suspects for a crime can: –Cooperate (stay silent, deny crime). If both cooperate, 1 year in jail. –Defect (confess). If both defect, 3 years (reduced since they confessed). –If A defects (blames the other), and B cooperate (silent) then A is free, and B serves a long sentence. CooperateDefect Cooperate -1, -1-5, 0 Defect 0, -5-3,-3

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35 Lecture Outline What is a game? –Few examples. Best responses Dominant strategies Nash Equilibrium –Pure –Mixed Existence and computation

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36 Notation We will denote a game G between two players (A and B) by G[ S A, S B, U A (a,b), U B (a,b)] where S A = set of strategies for player A (a S A ) S B = set of strategies for player B (b S B ) U A : S A x S B R (utility function for player A) U B : S A x S B R utility function for player B

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Normal-form game: Example Example : –Actions: S A = {“C”,”D”} S B = {“C”,”D} –Payoffs: u A (C,C) = -1, u A (C,D) = -5, u A (D,C) = 0, u A (D,D) = -3 CooperateDefect Cooperate -1, -1-5, 0 Defect 0, -5-3,-3

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A best response: intuition Can we predict how players behave in a game? First step, what will players do when they know the strategy of the other players? Intuitively: players will best-respond to the strategies of their opponents.

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39 A best response: Definition When player B plays b. A strategy a* is a best response to b if U A (a*,b) U A (a’,b) for all a’ S A (given that B plays b, no strategy gains A a higher payoff than a*)

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A best response: example Example: When row player plays Up, what is the best response of the column player? LeftRight Up 1,11,10,00,0 Bottom 0,00,01,11,1 LeftRight Up 1,11,10,00,0 Bottom 0,00,01,11,1

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Dominant Strategies (אסטרטגיות שולטות/דומיננטיות) Definition: action a* is a dominant strategy for player A if it is a best response to every action b of B. Namely, for every strategy b of B we have: U A (a*,b) U A (a’,b) for all a’ S A

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Dominant Strategies: in the prisoner’s dilemma CooperateDefect Cooperate -1, -1-5, 0 Defect 0, -5-3,-3 For each player: “Defect” is a best response to both “Cooperate” and “Defect. Here, “Defect” is a dominant strategy for both players…

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In the prisoner’s dilemma: (Defect, Defect) is a dominant-strategy equilibrium. Dominant Strategy equilibrium שווי משקל באסטרטגיות שולטות Definition: (a,b) is a dominant-strategy equilibrium if a is dominant for A and b is dominant for B. –(similar definition for more players) CooperateDefect Cooperate -1, -1-5, 0 Defect 0, -5-3,-3

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Dominant strategies: another example Who has a dominant strategy in this game? Dominant-strategy equilibrium? LeftmiddleRight Up 7,27,22,20,00,0 Bottom 3,43,45,20,40,4 We allowed ≥ in the definition. “Weakly dominant”

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Dominant strategies: pros and cons Plus: Strong solution. –Why should I play anything else if I have a dominant strategy? Main problem: Does not exist in many games…. LeftRight Up 1,11,10,00,0 Bottom 0,00,01,11,1

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46 Lecture Outline What is a game? –Few examples. Best responses Dominant strategies (golden balls)golden balls Nash Equilibrium –Pure –Mixed Existence and computation

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47 Nash Equilibrium How will players play when dominant-strategy equilibrium does not exist? –We will define a weaker equilibrium concept: Nash equilibrium A pair of strategies (a*,b*) is defined to be a Nash equilibrium if: a* is player A’s best response to b*, and b* is player B’s best response to a*.

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48 Nash Equilibrium: Definition A direct definition: A pair of strategies (a*,b*) is defined to be a Nash equilibrium if U A (a*,b*) U A (a’,b*) for all a’ S A U B (a*,b*) U b (a*,b’) for all b’ S B

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49 Nash Eq.: Interpretation No regret: Even if one player reveals his strategy, the other player cannot benefit. –this is not the case with non-equilibrium strategies Stability: Once we reach a Nash equilibrium, players have no incentive to alter their strategies. –Even after observing the strategies of the other players Necessary condition for an outcome chosen by rational players. –If players think that there is obvious outcome to the game, it must be a Nash equilibrium

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(Pure) Nash Equilibrium Examples: LeftRight Up 1,11,10,00,0 Bottom 0,00,01,11,1 SwerveStraight Swerve 0, 0-1, 1 Straight 1, -1-10,-10 Note: when column player plays “straight”, then “straight” is no longer a best response to the row player. Here, communication between players help.

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51 Nash vs. Dominant Strategies Every dominant strategy equilibrium is a Nash equilibrium. –If a strategy is a best response to all strategies of the other players, it is of course a best response to the dominant strategy of the other. The opposite is not true.

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Nash equilibrium: existence Does a Nash equilibrium always exist? –Note: we already saw that multiple equilibria are possible.

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Example 4: No (pure) Nash equilibrium. But how do people play this game? -1,11,-1 -1,1 TailHeads Tail Heads Matching Pennies (זוג או פרט) Is this an equilibrium?

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“Pure” Nash: pros and cons Good: –Describes “stable” outcomes. –May exist when dominant-strategy equilibria does not exist. –Simple and intuitive (especially when unique). Bad: –Not unique. What happens when multiple equilibria exist? –Does not always exist!

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55 Mixed strategies Consider the following strategy: “I will toss a coin. With probability ½ I will choose bottom. With probability ½ I will choose up.” If lottery is allowed, now each player has an infinite number of strategies… LeftRight Up 1,11,10,00,0 Bottom 0,00,01,11,1

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Mixed strategies: Definition Definition: a “mixed strategy” is a probability distribution over actions. –If {a 1,a 2,…,a m } are the pure strategies of A, then {p 1,…,p m } is a mixed strategy for A if -1,11,-1 -1,1 TailHeads Tail Heads 1/2 1/3 2/3 9/10 1/10 0 1 1/4 1/2 (1) (2) For all i

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57 Pure and Mixed strategies Clearly, every pure strategy is a mixed strategy as well. –That gives probability 1 to one of the pure strategies. We will simply use the term “strategies”.

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Expected payoff When the two players play mixed strategies, the payoff is the expected payoff. (הממוצע) LR T 3, -14, 24, 2 D 6, -51,91,9 2/3 1/3 3/4 1/4 What is the payoff of the row player? when the players play s A =(2/3, 1/3) and s B =(1/4,3/4) u A (s A,s B ) = 2/3 * ¼ * 3 + 1/3 * ¼ * 6 + 2/3 * ¾ * 4 + 1/3 * ¾ * 1 = 3.25

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Best response (w. mixed strategies) Definition: Consider a mixed strategy s B of player B. A strategy s * for player A is a best response to s B if no other pure strategy gains A higher expected payoff. Namely, –Note: we will later see that this implies that no mixed strategy is better for A than s*. U A (s*,s B ) U A (a’,s B ) for all a’ in S A

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Best response (w. mixed strategies) -1,11,-1 -1,1 זוגפרט זוג פרט 3/4 1/4 What is a best response to (1/4,3/4)? What would you do if you knew that your opponent plays one strategy more frequently? Will you play pure or mixed? 1 0

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Mixed strategies are realistic? Do people randomize? –Computers? Evolution? Stock markets? Teacher choosing questions in exams. Model long term behavior… Model uncertainty about the other players. זוג או פרט Basketball Soccer –How would you define strategy in penalty kicks? –“the player that kicks more often to the left”

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Nash eq. with mixed strategies Main idea: given a fixed behavior of the others, I will not change my strategy. Definition: (S A,S B ) are in Nash Equilibrium, if each strategy is a best response to the other. -1,11,-1 -1,1 זוגפרט זוג פרט 1/2

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Example: Battle of the Sexes Equilibria in “battle of the sexes”: –Two pure equilibria. –One mixed (2/3,1/3),(1/3,2/3) 2,12,10,00,0 0,00,01,21,2

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64 Lecture Outline What is a game? –Few examples. Best responses Dominant strategies Nash Equilibrium –Pure –Mixed Existence and computation

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65 Existence of equilibria Dominant strategies equilibria do not exist in every game. –Same goes for Pure Nash equilibria. What about Nash equilibria (with mixed strategies)? Good news: always exist.

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Nash’s Theorem Theorem (Nash, 1950): every game has at least one Nash equilibrium! –With some technical details about the set of strategies. –Proof uses fix-point theorems. Nash was awarded the Nobel prize for this work in 1994. Nash equilibrium (with mixed strategies): –Good: always exists. Models long term stability. –Bad: Less simple and intuitive. Multiple equilibria exist.

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67 Computing Equilibria Dominant strategy: for each player, check if she has a dominant strategy. Pure Nash: for each combination of actions, check if a player has a beneficial deviation. How can we find Nash equilibria in general? –This is a real problem in large games. Area of extensive research. –Easy in “small” games.

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68 Summary We learned about simultaneous-action games, represented by a matrix of payoffs. (Games in their “normal form”) –Next topic: sequential games. We wanted to predict the steady/stable state behavior on the games, and defined concepts of equilibria.

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69 Finding mixed equilibria We will use the following lemma: Lemma: let s A be a best response to s B. If s A chooses the pure strategies a, a’ with positive probability, then u A (a,s B )=u A (a’,s B ) Namely, if we sometime choose a and sometime choose a’, they gain us the same expected payoff (given a fixed behavior of the others).

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70 Proof of Lemma Assume that in best response: a is chosen with probability p a a’ is chosen with probability p a’ –p a,p a’ >0 Now if u A (a,s B ) > u A (a’,s B ), then this is not a best response: –The same strategy that chooses a with probability p a +p a’ and a’ with probability 0 gains A higher payoff. p a’ papa 0 p a + p a’ sBsB

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71 Finding mixed equilibria MS M 2,12,10,00,0 S 0,00,01,21,2 1/4 3/4 What is the best response to s B =(3/4,1/4)? Can it be s A =(½, ½)? u A (M,s B ) = ¾*2 + 1/4*0 = 1.5 u A (S,s B ) = ¾*0 + 1/4*1 = ¼ Expected payoff: ½*1.5 + ½*1/4 (1/2,1/2) cannot be a best response, niether (0.99,0.01) adding more mass to M will increase expected payoff of A. Again, here the best response is a pure strategy (“M”),

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72 Finding mixed equilibria So how can we find (strictly) mixed-strategy equilibria? We will use the lemma that we proved: if in equilibrium a player plays two pure strategies with positive probability, then the expected payoff from both strategies should be the same.

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73 Finding mixed equilibria MS M 2,12,10,00,0 S 0,00,01,21,2 1-q q Consider an equilibrium s A =(p,1-p), s B =(q,1-q) (q,p>0) then: u A (“M”,s B ) = u A (“S”,s B ) = If s A is a best response, we must have: u A (“M”,s B )=u A (“S”,s B ) that is : 2q = (1-q) q=1/3 Similarly, if s B is a best response then p=2/3. 1-p p q*2 + 0*(1-q) q*0 + (1-q)*1

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74 Finding mixed equilibria MS M 2,12,10,00,0 S 0,00,01,21,2 2/3 1/3 Note that since all mixed strategies are best response to s B =(1/3,2/3). But only s A =(2/3,1/3) ensures that s B =(1/3,2/3) is also a best response to s A. 1/3 2/3 u A (“M”,s B )=u A (“S”,s B )

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Equilibria All we said extends to more players: (s 1,…,s n ) is a Nash equilibrium, if for every i, s i is a best-response to the other strategies. (s 1,…,s n ) is a dominant-strategy equilibrium, if for every i, s i is the best response to any other set of strategies.

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Equilibria Take home message: Dominant-strategy equilibrium: my strategy is the best no matter what the others do. Exists in some games. Nash equilibrium: my strategy is the best given what the others are currently doing. Always exists.

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Example 1: coordination games LeftRight Left 1,11,10,00,0 Right 0,00,01,11,1 Row player שחקן השורות Column Player שחקן העמודות Right number: utility for Row Player Left number: utility for Column Player Without laws, when this game is repeated, what will happen?

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